Problem: $ A = \left[\begin{array}{rr}-1 & 5 \\ -1 & 1 \\ 3 & 2\end{array}\right]$ $ F = \left[\begin{array}{rr}5 & 3 \\ 1 & 5\end{array}\right]$ What is $ A F$ ?
Solution: Because $ A$ has dimensions $(3\times2)$ and $ F$ has dimensions $(2\times2)$ , the answer matrix will have dimensions $(3\times2)$ $ A F = \left[\begin{array}{rr}{-1} & {5} \\ {-1} & {1} \\ \color{gray}{3} & \color{gray}{2}\end{array}\right] \left[\begin{array}{rr}{5} & \color{#DF0030}{3} \\ {1} & \color{#DF0030}{5}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ A$ , with the corresponding elements in column $j$ of the second matrix, $ F$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ A$ with the first element in ${\text{column }1}$ of $ F$ , then multiply the second element in ${\text{row }1}$ of $ A$ with the second element in ${\text{column }1}$ of $ F$ , and so on. Add the products together. $ \left[\begin{array}{rr}{-1}\cdot{5}+{5}\cdot{1} & ? \\ ? & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ A$ with the corresponding elements in ${\text{column }1}$ of $ F$ and add the products together. $ \left[\begin{array}{rr}{-1}\cdot{5}+{5}\cdot{1} & ? \\ {-1}\cdot{5}+{1}\cdot{1} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ A$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ F$ and add the products together. $ \left[\begin{array}{rr}{-1}\cdot{5}+{5}\cdot{1} & {-1}\cdot\color{#DF0030}{3}+{5}\cdot\color{#DF0030}{5} \\ {-1}\cdot{5}+{1}\cdot{1} & ? \\ ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{-1}\cdot{5}+{5}\cdot{1} & {-1}\cdot\color{#DF0030}{3}+{5}\cdot\color{#DF0030}{5} \\ {-1}\cdot{5}+{1}\cdot{1} & {-1}\cdot\color{#DF0030}{3}+{1}\cdot\color{#DF0030}{5} \\ \color{gray}{3}\cdot{5}+\color{gray}{2}\cdot{1} & \color{gray}{3}\cdot\color{#DF0030}{3}+\color{gray}{2}\cdot\color{#DF0030}{5}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}0 & 22 \\ -4 & 2 \\ 17 & 19\end{array}\right] $